An oligopoly consists of n identical firms that produce a homogeneous product. The firm 1 chooses its output q1 to maximize its profits (All the firms do the same). So we have Π1 = p(Q).q1 – m.q1 where m is the constant marginal cost for each firm and p, the price, is a function of total output. By the identical firms assumption S1 = q1/Q = 1/n or Q=n.q1 (Equation 1) where S1 is the output share of firm 1. The first order condition is the following: (Equation 2) MR = p + q1.p´ = m where p´ is the derivative of price with respect to Q. The second order condition holds so we have profit maximization.
The Lerner Index (L) is the difference between the price and marginal cost as a function of price. It is also called price-cost margin or price-cost markup. So (Equation 3) L = (p – m) / p = - q1.p´ /p (By Equation 2)
The elasticity of demand is a characteristic of the demand curve and is defined as the percentage change in quantity that results from 1 percent change in price. If the elasticity of demand is very high (a large negative number), then the curve is said to be elastic. With a very elastic demand, a small price change induces a very large change in the quantity demanded. If the elasticity is low (a small negative number), the demand curve is inelastic, and a price change of 1 percent has relatively little effect on the quantity demanded. So the demand elasticity ε equals Q´.p/Q where Q´ is the derivative of total output with respect to p. So the reciprocal of the elasticity of demand 1/ε equals p´.Q/p (Equation 4)
So L equals - q1.Q.p´ / p.Q (By Equation 3 & multiply and divide with Q) = - (q1/Q) . (1/ ε) (By Equation 4) = -1/(n.ε) (By Equation 1) or –S1/ ε (By Equation 1)